A Lazer-Mckenna type problem with measures

Abstract

In this paper, we are concerned with a general singular Dirichlet boundary value problem whose model is the following: $$ \begin{cases} -\Delta u = \frac{\mu}{u^{\gamma}} & \text{in}\ \Omega,\\ u=0 &\text{on}\ \partial\Omega,\\ u>0 &\text{on}\ \Omega\,. \end{cases} $$ Here, $\mu$ is a nonnegative bounded Radon measure on a bounded open set $\Omega\subset\mathbb R^N$, and $\gamma>0$.